This is a demo for a type of animated visualization of 1 and 2 variable differential equations. Made initially for the 2024 Handmade Network Visibility Jam. Work in progress - more details to come when the project goes farther.
Streams archives from the jam:
Update at the conclusion of my jam time.
I added some command line options so that a user can choose what system they'd like to visualize and the parameters for it. So far I have 4 built in systems.
I also extended the system to support 2nd order equations. 1st order equations are equations where you directly specify the velocity. These make the nice uniform flow fields. 2nd order equations are equations where you directly specify the acceleration. In that sort of system the position of a point doesn't uniquely determine it's velocity, so the plot becomes more of a cluster of particles shooting around one another. You can still see things like, what states the system tends to be drawn towards, whether the system tends to form orbitals or shoot off to infinity, etc.
I haven't gotten the code in a state where I can share it all yet, but you can check out the plotter here for now: https://mr-4th.itch.io/delta-xy-plotter
I am two days of work on my jam project &delta-xy-plotter
I've got the particle system tuned to do a pretty good job accentuating the flow of the differential equation that is being plotted. It has rules for randomly spawning in new particles when possible, killing particles that are too close to standing still, and killing particles in areas that are too dense. This seems to create a nice balance of covering the plot in particles without making the density so high you can't actually "read" the plot at all.
In this animation I am using the classic Lotka-Volterra equations for modelling predator-prey interactions, and I am changing the base growth rates for each species over time to see the plot changing.
